"I didn't really bouce Eeyore. I had a cough and I happened to be behind Eeyore, and I said 'Grrr-oppp-ptschschschz'." -Tigger

According to author A. A. Milne, Eeyore was given to Christopher Robin Milne for Christmas, 1921. That would make Eeyore's official birthday: December 25, 1921. However, Eeyore's birthday in the book, and Disney's movie short, clearly does not take place in the winter. Therefore, Eeyore's book birthday is open to your interpretation.